CS: Blind Compressed Sensing, some more opaque lens and a comment

Here is a new entry from Arxiv where the authors want to make compressive sensing truly universal ... or blind in Blind Compressed Sensing by Sivan Gleichman, Yonina C. Eldar. The abstract reads:

The fundamental principle underlying compressed sensing is that a signal, which is sparse under some basis representation, can be recovered from a small number of linear measurements. However, prior knowledge of the sparsity basis is essential for the recovery process. This work introduces the concept of blind compressed sensing, which avoids the need to know the sparsity basis in both the sampling and the recovery process. We suggest three possible constraints on the sparsity basis that can be added to the problem in order to make its solution unique. For each constraint we prove conditions for uniqueness, and suggest a simple method to retrieve the solution. Under the uniqueness conditions, and as long as the signals are sparse enough, we demonstrate through simulations that without knowing the sparsity basis our methods can achieve results similar to those of standard compressed sensing, which relay on prior knowledge of the sparsity basis. This offers a general sampling and reconstruction system that fits all sparse signals, regardless of the sparsity basis, under the conditions and constraints presented in this work.

Sylvain Gigan pointed out to me the latest work on Opaque Lenses. The authors seem to now be able to perform some sort of superresolution in:

Optimal Concentration of Light in Turbid Materials by Elbert van Putten, Ad Lagendijk, Allard Mosk. The abstract reads:

In turbid materials it is impossible to concentrate light into a focus with conventional optics. Recently it has been shown that the intensity on a dyed probe inside a turbid material can be enhanced by spatially shaping the wave front of light before it enters a turbid medium. Here we show that this enhancement is due to concentration of light energy to a spot much smaller that a wavelength. We focus light on a dyed probe sphere that is hidden under an opaque layer. The light is optimally concentrated to a focus with an area that is for certain within 68% of the smallest focal area physically possible. A comparison between the intensity enhancements of both the emission and excitation light supports the conclusion of optimal light concentration.

Finally, Dick Gordon, one of the inventor of the ART algorithm sent me the following:

Steered microbeams and other active optics as compressive sensing
Dear Igor,
A search of Nuit Blanche archives on the words “steer” or “active” yielded surprisingly few relevant hits, except for the single pixel camera. I’ve listed a sampling of the active optics literature below. The sparse steered microbeam literature is reviewed in:

Richard Gordon (2010). Stop breast cancer now! Imagining imaging pathways towards search, destroy, cure and watchful waiting of premetastasis breast cancer [invited]. In: Breast Cancer - A Lobar Disease. T. Tot, Springer: in press.

The basic idea in applying a compressive sampling approach to these imaging problems is as follows. A detector or a radiation source/detector pair is aimed at the target. The data acquired is used to improve an estimate of the image. The image is evaluated for regions (in real or Fourier space) that are a) of interest, b) undersampled. Further data is acquired. Thus an iterative process occurs between the reconstructed image and data acquisition to improve it, in an image dependent manner. Greedy algorithm problems can be minimized by declaring any “uninteresting” region of at least temporary interest precisely for that reason, until proven indeed uninteresting. I’d like to suggest that the whole process has a lot in common with saccadic eye movements in building up understanding of a scene.

Let me give an example. A dark scene is illuminated one photon at a time, using turnstile photons that are emitted on command (Gordon, 2010). As the image is built up, a pattern recognition program starts to pick up interested features. Now these could be real, or due to random fluctuations. More photons are directed to these features. If they are real, their image is sharpened. If not, it is flattened. This nonlinear approach of active compressive sampling (ACS) may be the key to viewing a scene with the absolute minimum number of photons. For x-ray computed tomography (CT), ACS may be the key to orders of magnitude in dose reduction. Yours, -Dick Gordon gordonr@cc.umanitoba.ca

References:
Girkin, J.M., S. Poland & A.J. Wright (2009). Adaptive optics for deeper imaging of biological samples. Curr Opin Biotechnol 20(1), 106-110.

Hubin, N. & L. Noethe (1993). Active optics, adaptive optics, and laser guide stars. Science 262(5138), 1390-1394.

Hugot, E., M. Ferrari, G.R. Lemaitre, F. Madec, S. Vives, E. Chardin, D. Le Mignant & J.G. Cuby (2009). Active optics for high-dynamic variable curvature mirrors. Opt Lett 34(19), 3009-3011.

Lemaitre, G.R., P. Montiel, P. Joulie, K. Dohlen & P. Lanzoni (2005). Active optics and modified-Rumsey wide-field telescopes: MINITRUST demonstrators with vase- and tulip-form mirrors. Appl Opt 44(34), 7322-7332.

Macmynowski, D.G. (2009). Interaction matrix uncertainty in active (and adaptive) optics. Appl Opt 48(11), 2105-2114.

Signorato, R., O. Hignette & J. Goulon (1998). Multi-segmented piezoelectric mirrors as active/adaptive optics components. J Synchrotron Radiat 5(Pt 3), 797-800.

West, S.C. (2002). Interferometric Hartmann wave-front sensing for active optics at the 6.5-m conversion of the multiple mirror telescope. Appl Opt 41(19), 3781-3789.

Zhang, Y., D. Yang & X. Cui (2004). Measuring seeing with a Shack-Hartmann wave-front sensor during an active-optics experiment. Appl Opt 43(4), 729-734.

Three observations here:

• I am sure much can be done when steering is feasible, but what would probably be very interesting is to be able to use the surrounding medium as a way to do the focusing (see Wavefront Coding for Random Lens Imagers ? for instance).
• Adaptive Compressive Sensing is a subject that is being investigated that sometimes goes by the name of distilled sensing or simple adaptive compressive sensing. This is an area of much interest as results seem to show that adaptive solution seem to require fewer measurements at the price of an increased sampling complexity and attendant delays.
  
• With regards to X-rays, one would need to be able to do some steering of X-rays. Unless one is looking at soft X-rays for which some coating materials can provide some equivalent of mirror there not much one can do with radiation except shielding from them. What is more important is the lessons to learn for 40 years of coded aperture as explained by Gerry Skinner . The discussion is very interesting but it does not fit exactly with compressive sensing. Let us note that Gerry is actively involved in gamma lenses thereby avoiding coded aperture altogether (see Coded Mask Imagers: What are they good for ? The George Costanza "Do the Opposite" Sampling Scheme. )